CHAP. VII] PBIMmVE ROOTS, EXPONENTS, INDICES. 189 E. Desmarest37 devoted the last 86 pages of his book to primitive roots; the 70 pages claimed to be new might well have been reduced to five by the omission of trivial matters and the use of standard notations. To find (pp. 267-8) a primitive root of the prime P = 6g+ 1, where q is an odd prime, seek an odd solution of w2+3=0 (mod P) and set u = 2R 1; then R3z= 1 and R belongs to the exponent 6; thus we know the solutions of #6=1; let a be any integer prime to P and not such a solution; if a9=^lt then =±=a belongs to the exponent q, and ^aR is a primitive root of P; but, if a2fl^ 1, then a3fl== =F! (mod P), and ±a is a primitive root of P. If P=8Q+1 and Q are primes, then P=5 (mod 12) and 3 is a quadratic non-residue and hence a primitive root of P. Let P be a prime of the form 5q=± 2. Then w2= 5 (mod P) is not solvable. Thus, if a is a primitive root of P, 5= a', where e is odd. Thus if e is prime to P 1, 5 is a primitive root of P. It is recommended that 5 be the first number used hi seeking by trial a primitive root. And yet he announced the theorem (p. 283) that 5 is hi general a primitive root. If P is a prime 5# =±=2 also of the form 2nQ+l, where Q is an odd prime including 1, then (pp. 284^-6) 5 is a primitive root of P provided P is not a factor of S2" 1. He gave the factors of the latter and of 102" 1 f or n = 1, . . . , 5. Results, corresponding to those just quoted for 5, are stated for p = 7, 7, 10, 17. What is really given is a list of the linear forms of the primes P for which p is a quadratic non-residue. If, in addition, P = 2nQ+l, where Q is an odd prime, then p is a primitive root, provided p2"^! (mod P). The last condition is ignored in his statement of his results and again in his collection (pp. 297-8) of "principles which give primitive roots" entered in his table (pp. 298-300) giving a primitive root of each prime < 10000. V. A. Lebesgue38 proved that, if a and p = 2\z+l are primes, any quadratic non-residue x of p is a primitive root of p if J. P. Kulik39 gave for each prime p between 103 and 353 the indices and all the primitive roots of p. His manuscript extended to 1000. There is an initial table giving the least primitive root of the primes from 103 to 1009. G. Oltramare40 called x a root of order or index m of a prime piix belongs to the exponent (p \}/m modulo p. Let Xm(x) = 0 (mod p) be the congruence whose roots are exclusively the roots of order m of p. By changing x to xl/n, we obtain Xmn = <j> (x) = 0. If n^ , n2, . . . , n are the divisors > 1 of n, 87Th6orie des nombres. Traite" de 1'analyae inde*termine*e du second degre' 5, deux inconnuos suivi de 1'application de cette analyse & la recherche des racines primitives avec une table de ces racines pour tous les nombres premiers compris entre 1 et 10000, Paris, 1852, 308 pp. For errata, see Cunningham, Mess. Math., 33, 1903, 145. 38Nouv. Ann. Math., 11, 1852, 422HL 39Jour. fur Math., 45, 1853, 55-81. «°jm, 303-9.